If we assign each \(n\in\mathbb{N}\) a real number \(a_n\in\mathbb{R}\), we get a sequence \((a_n)\) with elements
\[(a_n) = a_1, a_2, a_3, \dots a_n\]
A sequence is a collection of numbers that follow a certain order or pattern. If a sequence starts at index \(k\), we call the sequence
\[(a_n)_{n\geq k} = a_k, a_{k+1}, a_{k+2}, \dots a_n = a_n - a_{k-1}\]
If \(n < \infty\), we call the sequence finite and otherwise infinite.
To define a sequence, we can state elements in explicit form
\[(a_n) = 1,2,3,4,\dots\]
or in closed form
\[(a_n) = n^2 = 1, 4, 9, 16, \dots\]
Additionally, we can define a sequence as a recurrence relation, where each term depends on the previous term:
\(a_1=0, a_2=a_1+1, a_3=a_2+2, a_4=a_3+3, \dots\)
A sequence is called a first-order recurrence if each element depends on only one previous term, such
\[(a_n) = \frac{1}{2}(a_{n-1} + \frac{c}{a_n})\]
A second-order recurrence for example is the famous Fibonacci sequence:
\[(a_n) = a_{n-1} + a_{n-2}\text{ with } a_0=a_1 = 1\]
Arithmetic Sequences
If the difference between two elements \(a_{n}\) and \(a_{n+1}\) is constant, such that
\[a_{n+1} - a_{n} = d\]
we call the sequence
\[(a_{n+1}) = a_{n} + d\]
an arthmetic sequence or arithmetic progression with \(a_1\) being the first term.
Arithmetic sequences can also be defined in a closed form:
\[(a_n) = a_1 + (n - 1)\cdot d\]
If \(d=0\) the sequence is constant. If \(d>0\), the sequence is monotonically increasing and if \(d<0\), the sequence is monotonically decreasing.
The name Aritmetic Sequence arises because each term other than the first is the arithmetic mean of its two neighbouring terms:
\[a_n = \frac{a_{n-1} + a_{n+1}}{2}\]
Partial Sum of an Arithmetic Sequence
Writing the partial sum \(S_n=a_1+a_2+\cdots+a_n\) once forward and once backward and adding the two line by line pairs up the terms:
\[ \begin{aligned} S_n &= a_1+a_2+\cdots+a_n\\ S_n &= a_n+a_{n-1}+\cdots+a_1\\ \hline 2S_n &= (a_1+a_n)+(a_2+a_{n-1})+\cdots+(a_n+a_1) \end{aligned} \]
Every one of the \(n\) pairs sums to the same value, since \(a_k+a_{n+1-k}=2a_1+(n-1)d=a_1+a_n\) for every \(k=1,\ldots,n\). So \(2S_n=n(a_1+a_n)\), giving the closed form
\[S_n=\frac{n}{2}(a_1+a_n)=n a_1+\frac{n(n-1)}{2}d.\]
Example. For the odd numbers \(1,3,5,\ldots\), \(a_1=1\) and \(d=2\), so the \(n\)-th term is \(a_n=2n-1\) and
\[1+3+5+\cdots+(2n-1)=\frac{n}{2}\bigl(1+(2n-1)\bigr)=\frac{n}{2}\cdot 2n=n^2.\]
This identity also has a direct visual proof. Arrange \(n^2\) dots in an \(n\times n\) grid, and group them into \(n\) nested "L"-shaped layers, where layer \(k\) (counting from \(0\)) consists of every dot in row \(k\) or column \(k\) that has not already been counted in an earlier layer:
Layer \(k\) has \(2k+1\) dots (\(k\) dots finishing off the row above it, \(k\) dots finishing off the column to its left, and the one corner dot where row and column meet), so the layers have sizes \(1,3,5,7,9,\ldots\) for \(k=0,1,2,3,4,\ldots\). Since the whole \(n\times n\) grid is partitioned into exactly these \(n\) layers, summing the layer sizes gives \(1+3+5+\cdots+(2n-1)=n^2\) directly, without needing the pairing argument at all.
Geometric Sequences
If the quotient \(q\) of two consecutive elements is constant, such that
\[\frac{a_{n+1}}{a_n} = q\]
we call the sequence
\[(a_{n+1}) = a_n\cdot q\]
a geometric squence with \(a_1\) being the first term.
Geometric squences can also be defined in a closed form:
\[(a_n) = a_1\cdot q^{n-1}\]
Geometric series have the following behavior:
- If \(0<q<1\), the sequence is monotonically decreasing and converges to zero as \(q^n\to 0\) as \(n\to\infty\).
- If \(-1<q<0\), the sequence is alternating in sign and converges to zero as \(q^n\to 0\) as \(n\to\infty\).
- If \(q>1\), the sequence is monotonically increasing as \(q^n\to\infty\) as \(n\to\infty\).
- If \(q<-1\), the sequence is unbound and alternates in sign.
- If \(q=0\) the sequence is constant \(0\).
- If \(q=1\) the sequence is constant \(a_1\).
- If \(q=-1\) the sequence is alternating between \(a_1\) and \(-a_1\).
The name Geometric Sequence arises because each term other than the first is the geometric mean of its two neighbouring terms:
\[|a_n| = \sqrt{a_{n-1} \cdot a_{n+1}}\]
Linear Recurrence Sequences
In practice often a combination of arithmetic and geometric sequences aries. If we have a constant \(d\) and a constant \(q\) we call the sequence
\[(a_{n+1}) = a_n \cdot q + d\]
a linear recurrence sequence with \(a_1\) being the first term.
Linear recurrence sequences can be expressed in a closed form as
\[\begin{array}{rl} a_2 &= a_1 \cdot q + d\\ a_3 &= (a_1 \cdot q + d) \cdot q + d = a_1 \cdot q^2 + d\cdot q + d\\ a_4 &= (a_1 \cdot q^2 + d\cdot q + d) \cdot q + d = a_1 \cdot q^3 + d\cdot q^2 + d\cdot q + d\\ & \vdots\\ a_n &= a_1\cdot q^{n-1} + d\cdot(q^{n-2} + \dots + q + 1)\\ &= \begin{cases} a_1\cdot q^{n-1} + d\cdot\left(\frac{q^{n-1}-1}{q-1}\right) & \text{for } q \neq 1 \\ a_1 + d \cdot (n - 1) & \text{for } q = 1 \end{cases} \end{array}\]
Note that arithmetic sequences are linear recurrence sequence with \(q=1\) and geometric sequenecs are linear recurrence sequences with \(d=0\).
Inferring Parameters for Linear Recurrence Sequences
Let’s say we get the sequence \(a_1, a_2, a_3, \dots\), how can we infer the parameter \(q\) and \(d\) of this sequence?
We know that
\[\begin{array}{rl} a_2 &= a_1 \cdot q + d\\ a_3 &= a_2 \cdot q + d \end{array}\]
Subtracting both equations from another gives
\[a_3 - a_2 = (a_2 - a_1)\cdot q\]
From which follows that
\[q = \frac{a_3 - a_2}{a_2 - a_1}\]
Taking the first equation again gives
\[d = a_2 - a_1 \cdot q\]
Monotonicity of Sequences
An important criterion to distinguish sequences is the monotonicity of sequences. If the elements get larger with increasing index, the sequence is called monotonically increasing, if the elements get smaller with increasing index, the sequence is called monotonically decreasing. If the sequence is changing the sign, we call it an alternating sequence.
For a sequnce \((a_n)\) we say
\[\begin{array}{l} a_{n+1}\geq a_n \,\forall n\Leftrightarrow (a_n)\text{ is monotonically increasing}\\ a_{n+1}\leq a_n \,\forall n\Leftrightarrow (a_n)\text{ is monotonically decreasing}\\ a_{n+1}= a_n \,\forall n\Leftrightarrow (a_n)\text{ is constant}\\ \end{array}\]
If the relations hold for \(>\) and \(<\) instead of \(\geq\) and \(\leq\), we call the sequences strictly monotonically increasing and decreasing respectively.
Boundedness of Sequences
A sequence \((a_n)\) is called bounded if there exist two real numbers \(s\) and \(S\) such that all elements of the sequence are between lower bound \(s\) and upper bound \(S\):
\[s\leq a_n\leq S\,\forall n\]
Limits of Sequences
A sequence \((a_n)\) has a limit \(a\) if an infinite neighborhood of \(a<\infty\) results in a finite set of elements. In this case we say
\[\lim\limits_{n\to\infty} a_n=a\]
If a series has a limit, we call it convergent, otherwise divergent.
We can say the following about sequences:
- A limit is always distinct.
- \(\pm\infty\) is no valid limit
- Every monotonical and bounded sequence is convergent.
- Every convergent sequence is bounded.
- The converse of the previous point fails: a bounded sequence need not be convergent, such as the alternating sequence \(a_n=(-1)^n\), which is bounded between \(-1\) and \(1\) but never settles on a single limit.
- Every bounded sequence still has a convergent subsequence, a fact known as the Bolzano–Weierstrass theorem. For \(a_n=(-1)^n\) above, the subsequence of even indices is constantly \(1\) and the subsequence of odd indices is constantly \(-1\), each trivially convergent even though the full sequence is not.
Sequences that converge to zero are called null sequence.
We can say that a sequence \((a_n)\) converges to \(a\) if the sequence \((a_n-a)\) is a null sequence.
Cauchy convergence criterion
Augustin-Louis Cauchy formulated a more general criterion for the case that the limit \(a\) is not known beforehand. It says that a sequence \((a_n)\) is convergent iff for every arbitrary small \(\epsilon>0\) exists a \(n_0\in\mathbb{N}\) such that all all sequence elements \(a_i, a_j\) follow \(|a_i-a_j|<\epsilon\), begining with element \(a_{n_0}\). More formally this means
\[\forall\epsilon>0\exists n_0\in\mathbb{N}\forall i,j\geq n_0 : |a_i-a_j|<\epsilon\]
Important sequence limits
\[\begin{array}{rl} \lim\limits_{n\to\infty}\frac{a}{n} &= 0\forall a\in\mathbb{R}\\ \lim\limits_{n\to\infty}a^n &= 0\,\forall |a|<1\\ \lim\limits_{n\to\infty}\sqrt[n]{a}&=1\,\forall a\in\mathbb{R}^+\\ \lim\limits_{n\to\infty}\sqrt[n]{n} &= 1\\ \lim\limits_{n\to\infty}\left(1+\frac{1}{n}\right)^n &= e\\ \lim\limits_{n\to\infty}\left(1-\frac{1}{n}\right)^n &= \frac{1}{e}\\ \lim\limits_{n\to\infty}\left(1+\frac{k}{n}\right)^n &= e^k\\ \end{array}\]
Where \(e=2.71828\dots\) is the irrational number called Euler’s number.
The first two of these, that \(a^n\to0\) for \(|a|<1\) and that \(\sqrt[n]{n}\to1\), are not obvious from the algebra of limits alone; both are proved from Bernoulli's inequality.
Discrete Derivative of Sequences
If a sequence is interpreted as sampled values \((a_t)=f(t)\) over time, the derivative idea is represented by a finite difference:
\[ f'(t)\approx \frac{a_{t+1}-a_t}{\Delta t}. \]
For unit time steps \(\Delta t=1\), this simplifies to \(f'(t)\approx a_{t+1}-a_t\). This is the standard discrete slope used for sampled measurements.
For an arithmetic sequence \(a_n=a_1+d(n-1)\), we have \(a_{n+1}=a_1+dn\), hence
\[ \frac{a_{n+1}-a_n}{1} = \bigl(a_1+dn\bigr)-\bigl(a_1+d(n-1)\bigr) = d. \]
So the discrete derivative of an arithmetic sequence is the constant difference \(d\).